Sunday, January 22, 2006

Misleading statistics: second-order ratios

I saw an ad on TV the other day, promising that its product would "kill 99% of most household germs", or something like that. Of course, "most" could mean anything over 50%, and to take a proportion (even 99%) of that will make it even less, so the promise doesn't really say much. It's almost as bad as the sales where everything is "up to 50% off" (which of course is consistent with most of the discounts actually being far less). Damn advertisers.

More generally though, it can be difficult to report second-order percentages in a non-misleading way. Suppose a political candidate starts with 10% of the vote. If someone reported that his popularity had "increased by 50%", you might be startled into thinking that he was now winning the race, with 60% of the vote. But they might instead merely mean to say that he increased his vote share by five percentage points, to a total of 15% -- which is, after all, half again as good as (hence, 50% improvement on) his original position of 10%.

Anyway, it would be nice if people took more care to resolve this ambiguity when reporting these sorts of statistics. Clarity might be achieved by describing the first sort of increase as "an improvement of 50 percentage points", which I think is more clearly talking about the first-order units. To report second-order percentages, one might specify the background set, e.g. "50% of his original 10 per cent share".

Such care might significantly improve 99% of some of the world.


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5 comments:

  1. You are not supposed to be misleading to an average person. You can't hide behind "well i was technically correct" in a "I did not have sexual relations with that woman" sort of a way.

    But we dont even enforce the laws (but I think we should) against lies (/totally unsubstantiated facts) usually so it is a little academic.

    Anyway - I wonder about "blackcurrents have 4 times the vitamin C of oranges"
    A) they are tiny obviously that would be impossible
    B) if the juice has more per volume then they should say that AND that is irelevant because they water it down in their drinks and orange juice may or may not be watered down.

    I guess what protects them is that average people dont trust advertising anymore.

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  2. oh yeah - and I understand vitamin C decays with time in sunlight....

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  3. Richard,

    One way of reading the cleaning product's claim is that the 99% applies to the token microorganisms it encounters, and the 'most' applies to the types of germ. So for most kinds of household germs, this product will kill 99% of the tokens.
    You're right that it would be good to take care of such ambiguity, but reporting involving statistics is so bad in so many ways already. You almost never hear about the methodology of the study, without which the statistics are virtually meaningless. And apparently reporters don't even consider the mutual consistency of the statistics they report.

    (See here for a study that produced such mutually incongruous statistics as:

    80% of Americans think "The U.S. government is hiding the fact that it knows of the existence of aliens."

    and

    64% "said that aliens have contacted humans."

    yet only

    "54 percent believe intelligent life exists outside Earth.")

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  4. Obviously, there are stupid extraterrestrial life forms that the US government knows about even though these aliens have not contacted us, and the government is doing a bad job of hiding these facts from the public, as nearly one in five Americans is already aware of the truth. Many ufologists are half-aware that a substantial portion of the majority of skeptics can still grasp about 40% of some aspects of the truth about a few of our alien visitors, if they'd just pay attention to the signs.

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  5. Derek, yeah, I thought of the type/token thing just after posting, which would at least make the ad less gratuitously incomprehensible. Though still unfortunately misleading, I think.

    And yes, there are many worse (ab)uses of statistics around. Still, that's no reason for me to not complain about this one ;-)

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